\chapter{Simplicial Sets}
\noindent \tcb{Motivation to study simplicial sets.} The equivalence between simplicial sets and CW complexes...the link between combinatorics and topology...the introcudtion of invariants under homotopy equivalence)
\section{Triangulated Spaces}\label{TriangulatedSpaces}
\noindent We define $n$-dimensional simplex to be the sub-topological space of $\RR^n$, determined by the set
$$\Delta_n=\{(t_0,t_1,...,t_n)\in \RR^{n+1}|\displaystyle\sum_{i=0}^n t_i=1,t_i\geq0 \}$$
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw [black,->] (0,0) -- (0,2);
\draw [black,->] (0,0) -- ( 1.0000,-1.7321);
\draw [black,->] (0,0) -- ( 1.7321 ,1.0000);
\draw [black, fill=gray] (0,1) -- (.5,- 0.8660) -- ( 0.8660 ,0.5) -- (0,1);
\node at (-.2,1) {$v_3$};
\node at (.4,- 0.9660) {$v_1$};
\node at ( 0.9660 ,0.6) {$v_2$};
\node at (-.2,2) {$t_3$};
\node at (.9, -1.8321) {$t_1$};
\node at ( 1.8321 ,1.1) {$t_2$};
\node at (-.2,0) {$0$};
%\draw[green, ultra thick, domain=0:0.5] plot (\x, {0.025+\x+\x*\x});
\end{tikzpicture}
%\begin{tikzpicture}
%\draw [black, ultra thick, fill=gray] (0,1) -- (.5,- 0.8660) -- ( 0.8660 ,0.5) -- (0,1);
%\end{tikzpicture}
\caption{$\Delta_2$}
\end{center}
\end{figure}
\noindent For every $I\subseteq [n]$, we define the $I^{th}$-face of $\Delta_n$ to be $\{(t_0,t_1,...,t_n)\in \Delta_n|t_i=0\text{ for }i\nin I\}$.
\noindent For every increasing map $\nu:[m]\rightarrow [n]$, we define:
$$
\begin{array}{lcccl}
\Delta_{\nu}:&\Delta_m&\rightarrow&\Delta_n\\
&u&\mapsto&t\ ,&t_i=\left\{\begin{array}{lr}
u_{{\nu}^{-1}(i)}& i\in \Imm \nu\\
0& \text{otherwise}
\end{array}\right.
\end{array}
$$
\noindent We distinguish points $v_i=(0,0,..,1,..,0)\in\Delta_n$, zeros apart from the $i^{th}$-coordinate,for $i=0..n$, and we call them the vertices of the simplex. Notice that $\Delta_{\nu}$ maps injectively $v_j$ to $v_{\nu(j)}$. For every such $\nu$, we understand $\Delta_{\nu}$ as an identification of a $m$-dimensional simplex with a face of $n$-dimensional simplex, namely $\Imm\ \Delta_{\nu}$, i.e. an embedding of $\Delta_m$ on the $\Imm \ \nu$-face of $\Delta_n$.\\
\noindent We define gluing data of simplices to be a contra-variant functor $$X:\Simp_{inc}^{op}\rightarrow\Sets,$$
\noindent where $\Simp_{inc}$ is the category of finite sets and increasing maps between them.\\
\noindent The of elements of $X_{(n)}:=X([n])$ is understood to be the set of $n$-dimensional simplices to be glued, for every increasing $\nu:[m]\rightarrow [n]$, and $X(\nu)$ specify which $\Delta_m$ to be identify with the $\Imm \nu$-face of each of the $\Delta_n$.\\
\noindent The requirement of the functoriality of  $X$ means that $X(id_{[n]})=id_{X_{(n)}}$, i.e. different elements of $X_{(n)}$ are understood to represent different $n$-dimensional simplices, so that the cardinality of $X_{(n)}$ \tcb{is minimal}. and $X(\nu \mu)=X(\mu) X(\nu)$, so that the $\Imm \mu\!-\!$face of $\Imm\nu\!-\!$face of a simplex is a face, namely the $\Imm \nu\mu\!-\!$face.\\

\noindent For every gluing data $X$, we define the topological space $$\mid X \mid=\left(\displaystyle\bigsqcup_{n=0}^{\infty}(\Delta_n\times X_{(n),Dis})\right)/R$$
 where $X_{(n),Dis}$ s the discrete topogicpace with the underlying set $X_{(n)}$ $R$ is the smallest equivalence relation that identify $(u,y)\in\Delta_m\times X_{(m),Dis}$ and $(t,x)\in\Delta_n\times X_{(n),Dis}$, such that there is a increasing map $\nu:[m]\rightarrow [n]$ and 
$$y=X(\nu)(x),\ and \  t=\Delta_{\nu} (u)$$.
\noindent $\mid X \mid$ is topologised with the weakest topology that makes the canonical map 
\begin{equation}\label{eq_TriRealisationTop}
\Pi_X:\displaystyle\bigsqcup_{n=0}^{\infty}(\Delta_n\times X_{(n),Dis})\rightarrow \mid X \mid
\end{equation}
continuous, we call this topology the canonical topology on $\mid X \mid$.\\
\noindent We call the gluing data a triangulation of $\mid X \mid$, and we call the induced topological space with the triangulation a triangulated space.
\begin{definition}[Triangulable Spaces]
\tcr{Let $(X,\tau)$ be a topological space, we say that $X$ is triangulable if it is homeomorphic to a triangulated space}.
\end{definition}
\noindent \tcb{Give an example of a topological space that is not triangulated}.\\
\begin{remark}
\noindent In the previous construction, we do not identify two proper faces of two simplices directly, but we rather identify each of them with one simplex of the same  dimension of the common face.
\end{remark}
\begin{example}[$S^2$] Let $X$ be a gluing data given by $X_{(0)}=\{P_1,P_2,P_3\},X_{(1)}=\{L_1,L_2,L_3\},X_{(2)}=\{F_1,F_2\}$, and $X_{(n)}=\emptyset$ otherwise, and the maps:\\


{$\begin{array}{ll}
{\nu}:[0]\rightarrow[1]$, ${\nu}(0)=0:&X({\nu})(L_1)=P_1,X({\nu})(L_2)=P_2,X({\nu})(L_3)=P_2\\
{\nu}:[0]\rightarrow[1]$, ${\nu}(0)=1:&X({\nu})(L_1)=P_2,X({\nu})(L_2)=P_2,X({\nu})(L_3)=P_3\\
\\
{\nu}:[1]\rightarrow[2]$, ${\nu}(0)=0,{\nu}(1)=1:&X({\nu})(F_1)=L_1,X({\nu})(F_2)=L_3\\
{\nu}:[1]\rightarrow[2]$, ${\nu}(0)=0,{\nu}(1)=2:&X({\nu})(F_1)=L_2,X({\nu})(F_2)=L_2\\
{\nu}:[1]\rightarrow[2]$, ${\nu}(0)=1,{\nu}(1)=2:& X({\nu})(F_1)=L_1,X({\nu})(F_2)=L_3\\
\end{array}
$}
\begin{tikzpicture}[scale=1]
\draw[blue,very thick] (0cm,0cm) -- (.5cm,0.86603cm);
\draw[blue,very thick]  (.5cm,0.86603cm) -- (1cm,0cm);
\draw[red,very thick] (1cm,0cm) -- (0cm,0cm);

\draw[green,very thick] (0cm,-.1cm) -- (.5cm,-0.96603cm);
\draw[green,very thick]  (.5cm,-0.96603cm) -- (1cm,-.1cm);
\draw[red,very thick] (1cm,-.1cm) -- (0cm,-.1cm);
\end{tikzpicture}\\

\noindent and identities and compositions otherwise. Then, we can see readily that the triangulated space $\mid X \mid$ is given by identifying the lines of the same colors in the figure, identifying the end points of the red line. Then, we can see that $\mid X \mid$ is homeomorphic to the sphere $S^2$.
\end{example}
\begin{remark}
\noindent Note that the same space is homeomorphic to the topological realisation given by the triangulation $X$, such that $X_{(0)}=\{P_1,P_2,P_3\},X_{(1)}=\{L_1,L_2,L_3\},X_{(2)}=\{F_1,F_2\}$, and $X_{(n)}=\emptyset$ otherwise, and the maps:\\

\noindent $\begin{array}{ll}
{\nu}:[0]\rightarrow[1]$, ${\nu}(0)=0:&X({\nu})(L_1)=P_1,X({\nu})(L_2)=P_2,X({\nu})(L_3)=P_3\\
{\nu}:[0]\rightarrow[1]$, ${\nu}(0)=1:&X({\nu})(L_1)=P_2,X({\nu})(L_2)=P_3,X({\nu})(L_3)=P_1\\
\\
{\nu}:[1]\rightarrow[2]$, ${\nu}(0)=0,{\nu}(1)=1:&X({\nu})(F_1)=L_1,X({\nu})(F_2)=L_1\\
{\nu}:[1]\rightarrow[2]$, ${\nu}(0)=0,{\nu}(1)=2:&X({\nu})(F_1)=L_3,X({\nu})(F_2)=L_3\\
{\nu}:[1]\rightarrow[2]$, ${\nu}(0)=1,{\nu}(1)=2:& X({\nu})(F_1)=L_2,X({\nu})(F_2)=L_2\\
\end{array}
$
\begin{tikzpicture}[scale=1]
\draw[blue,very thick] (0cm,-1cm) -- (.5cm,-0.13397cm);
%\draw[red,very thick]  (.5cm,0.86603cm) -- (1cm,0cm);
\draw[green,very thick]  (.5cm,-0.13397cm) -- (1cm,-1cm);
\draw[red,very thick] (1cm,-1cm) -- (0cm,-1cm);

\draw[blue,very thick] (0cm,-1.1cm) -- (.5cm,-1.96603cm);
\draw[green,very thick]  (.5cm,-1.96603cm) -- (1cm,-1.1cm);
\draw[red,very thick] (1cm,-1.1cm) -- (0cm,-1.1cm);
\end{tikzpicture}\\

\noindent and identities and compositions otherwise. Then, we can see readily that the triangulated space $\mid X \mid$ is given by identifying the lines of the same colors in the figure, which is also homeomorphic to the sphere $S^2$.
\end{remark}
\section{Simplicial Category}
\noindent \tcb{Type motivation, In this section we follow May, Gelfand and manin, Hovey, Jardine, and Palacios notes} \\
% In this section we generalise the the above notion of triangulation as follows:\\
%\tcr{Why to study simplicial sets}\\

\noindent Consider the simplicial category $\Simp$, i.e. the category of finite sets and order-preserving maps between them (non-decreasing maps). Then, we distinguish the below sets of morphisms:\\
For each $n\geq 1$, $0\leq i \leq n$, the $i^{th}$-face, $\partial_n^i:[n-1]\rightarrow [n]$, skipping the value $i$, given by $$\partial_n^i(j)=\left\{\begin{array}{lr}
j& j<i\\
j+1& j\geq i
\end{array}\right.$$
For each $n\geq 0$, $0\leq i \leq n$, the $i^{th}$-degenerations, $\sigma_n^i:[n+1]\rightarrow [n]$, repeating the the value $i$, given by $$\sigma_n^i(j)=\left\{\begin{array}{lr}
j& j\leq i\\
j-1& j>i
\end{array}\right.$$
Then, it is readily seen that these morphisms satisfies the relations:
\begin{equation}\label{eq_SimpRel}
\begin{array}{ll}
\partial_{n+1}^j\partial_n^i=\partial_{n+1}^i\partial_n^{j-1}&\text{for }i<j\\
\sigma_{n}^j\sigma_{n+1}^i=\sigma_{n}^i\sigma_{n+1}^{j+1}&\text{for }i\leq j\\
\sigma_{n-1}^j \partial_n^i=\left\{
\begin{array}{l}
\partial_{n-1}^i\sigma_{n-2}^{j-1} \\
id_{[n-1]}\\
\partial_{n-1}^{i-1}\sigma_{n-2}^{j}
\end{array}\right.&\!\!\!
\begin{array}{l}
\text{for } i<j\\
\text{for } i=j \text{ or } i=j+1\\
\text{for } i>j+1\\
\end{array}
\end{array}
\end{equation}
We have the flowing diagram of faces and degenerations:
\begin{equation}\label{eq_SimpDiagram}
\xymatrix{
[0]\ar@<+10pt>[rr]^{\partial_1^0}\ar@<-10pt>[rr]_{\partial_1^1}&&[1]\ar[ll]|{\sigma_0^0}\ar[rr]|{\partial_2^1}\ar@<+20pt>[rr]^{\partial_2^0}\ar@<-20pt>[rr]_{\partial_2^2}&&[2]\ar@<+10pt>[ll]|{\sigma_1^0}\ar@<-10pt>[ll]|{\sigma_1^1}
\ar@<+30pt>[rr]^{\partial_3^0}\ar@<+10pt>[rr]|{\partial_3^1}\ar@<-10pt>[rr]|{\partial_3^2}\ar@<-30pt>[rr]_{\partial_3^3}&&[3]
\ar[ll]|{\sigma_2^1}\ar@<+20pt>[ll]|{\sigma_2^0}\ar@<-20pt>[ll]_{\sigma_2^2}...}
\end{equation}

The importance of these morphisms that they generate all the morphisms of the category $\Simp$, as shown in the below lemma:
\begin{lemma}\label{SimpRelFactorisation}
Any morphism $\mu:[m]\rightarrow [n]$ of $\Simp$ is a composition of faces and degenerations, and can be written uniquely:
$$\mu=\partial_n^{i_1}\partial_{n-1}^{i_2}...\partial^{i_s}_{n-s+1}\sigma^{j_t}_{m-t}\sigma^{j_{t-1}}_{m-t+1}...\sigma^{j_1}_{m-1}$$
where $n-s=m-t$, $n\geq i_1> i_2>...> i_s\geq 0$ and $m-1\geq j_1>j_2>...>j_t\geq 0$.
\end{lemma}
\begin{proof}[Idea of the proof]
\tcr{\cite[P15]{GelMan03}}. It can be proven by induction on the both the number of skipped values, and the number of the repeated valued in the codomain.\\
\end{proof}
\noindent Below is an illustrated example:
$$
\xymatrix @C=1pc @R=.25pc
{
				&\cdot&&	&&				&				&				&\cdot\\
\cdot \ar[ru]	&\cdot&&	&&\cdot\ar[rd]	&				&\cdot\ar[ru]	&\cdot\\
\cdot \ar[r]	&\cdot&&=	&&\cdot\ar[rd]	&\cdot\ar[ru]	& \cdot\ar[r]	&\cdot\\
\cdot \ar[ru]	&\cdot&& 	&&\cdot\ar[r]	&\cdot\ar[ru]	&\cdot\ar[r]	&\cdot\\
\\
[2]\ar[r]_{\mu}&[3]&&=&&[2]\ar[r]_{\sigma_1^0}&[1]\ar[r]_{\partial_2^0}&[2]\ar[r]_{\partial_3^2}&[3]
}
$$ 
\begin{remark}
Avoid the confusion might arise form the name of the superscript of faces, $\partial_1^0$ does not map $0$ to $0$, it rather skips $0$ and maps $0$ to $0$, and so on.
\end{remark}
\section{(Co)Simplicial Objects}
\noindent Hereby, we recall the notion of simplicial and cosimplicial objects in given categories. With comparison to triangulated spaces, cosimplicial objects provide the "blocks" of realisation of simplicial objects, whereas the simplicial objects play the role of providing the gluing data of these realisation blocks. \tcb{This will be explained later after providing the definitions and concrete examples}
\begin{definition}
Let $\bcC$ and $\bcD$ be categories, then we define the simplicial objects of $\bcC$ to be contra-variant functors:
$$X:\Simp^{op}\rightarrow \bcC$$
and we define cosimplicial objects of $\bcD$ to be covariant functors:
$$R:\Simp\rightarrow \bcD$$
\end{definition}
\noindent We write $X_n$, $R_r$ instead of $X([n])$ and $R([n])$, respectively. Also, when it is not ambiguous, we will use the short notation:\\
$$d_i^n:=X(\partial_n^i),\ s_i^n:=X(\sigma_n^i) \text{ and } d^i_n:=R(\partial_n^i), \ s_n^i:=R(\sigma_n^i)$$
Then, \eqref{eq_SimpRel} induce the same relations on $d_n^i,s_n^i$, and the below corresponding relations on $d_i^n, s_i^n$:
\begin{equation}\label{eq_SimpObRel}
\begin{array}{ll}
d^n_i d^{n+1}_j=d^n_{j-1} d^{n+1}_i&\text{for }i<j\\
s^{n+1}_i s^{n}_j=s^{n+1}_{j+1} s^{n}_i&\text{for }i\leq j\\
d^n_i s^{n-1}_j =\left\{
\begin{array}{l}
s^{n-2}_{j-1} d^{n-1}_i \\
id^{X_{n-1]}}\\
s^{n-2}_{j} d^{n-1}_{i-1}
\end{array}\right.&\!\!\!
\begin{array}{l}
\text{for } i<j\\
\text{for } i=j \text{ or } i=j+1\\
\text{for } i>j+1\\
\end{array}
\end{array}
\end{equation}
Then, for the cosimplicial objects $R$, we have a similar diagram of \eqref{eq_SimpDiagram}, substituting $[n],\partial_n^i,$ and $\sigma_n^i$ with $R_n,d_n^i,$ and $s_n^i$, respectively. Whereas, for the simplicial object $X$ , \eqref{eq_SimpDiagram} induces the flowing diagram:
\begin{equation}\label{eq_SimpObDiagram}
\xymatrix{
X_0
\ar[rr]|{s^0_0}&&
X_1
\ar@<+10pt>[ll]^{d^1_0}\ar@<-10pt>[ll]_{d^1_1}
\ar@<+10pt>[rr]|{s^1_0}\ar@<-10pt>[rr]|{s^1_1}&&
X_2
\ar[ll]|{d^2_1}\ar@<+20pt>[ll]^{d^2_0}\ar@<-20pt>[ll]_{d^2_2}
\ar[rr]|{s^2_1}\ar@<+20pt>[rr]|{s^2_0}\ar@<-20pt>[rr]_{s^2_2}&&
X_3
\ar@<+30pt>[ll]^{d^3_0}\ar@<+10pt>[ll]|{d^3_1}\ar@<-10pt>[ll]|{d^3_2}\ar@<-30pt>[ll]_{d^3_3}
...}
\end{equation}
One can think of the simplicial object $X$ of the category $\bcC$ as a set of objects of $\bcC$, namely $X_n$ with the morphisms $d_i^n,s_i^n$, satisfying \eqref{eq_SimpObRel}, and all their possible composition. Whereas cosimplicial object $R$ of the category $\bcD$ can be thought of a set of objects of $\bcD$, $R_n$ with the morphisms $d^i_n,s^i_n$, satisfying equations similar to \eqref{eq_SimpRel}, and all their possible composition.

\noindent We define the category of simplicial objects in $\bcC$ to be the functor category $\bcC^{\Simp^{op}}$, and the  category of cosimplicial objects in $\bcD$ to be the functor category $\bcD^{\Simp}$, i.e. the morphisms of (co)simplicial objects are natural transformations. Hence, the morphism of simplicial objects in $\bcC$, $f:X\rightarrow Y$  is a set of morphisms $f_n$ in $\bcC$ that makes the below diagram commutes:
\begin{equation}\label{eq_SimpObMor}
\xymatrix{
X_n\ar[r]^{X_{\mu}}\ar[d]_{f_n}&X_{m}\ar[d]^{f_{m}}\\
Y_n\ar[r]_{Y_{\mu}}&Y_{m}
}
\end{equation}
for any order preserving map $\mu:[m]\rightarrow [n]$, particularly, the following diagrams commute:
$$
\xymatrix{
X_n\ar[r]^{d_i^n}\ar[d]_{f_n}&X_{n-1}\ar[d]^{f_{n-1}}\\
Y_n\ar[r]_{d_i^n}&Y_{n-1}
}\ \ \ 
\xymatrix{
X_n\ar[r]^{s_i^n}\ar[d]_{f_n}&X_{n+1}\ar[d]^{f_{n+1}}\\
Y_n\ar[r]_{s_i^n}&Y_{n+1}
}
$$
We denote the category of simplicial objects of $\bcC$ by $\Simp^{op}\bcC$, and the category of cosimplicial objects of $\bcD$ by $\Simp\bcD$.\\

\noindent Consider have the functor $(-)_n:\Simp^{op}\bcC\rightarrow \bcC$ that sends $X$ to $X_n$, and $f$ to $f_n$. Then, for every functor $F:J\rightarrow \Simp^{op}\bcC$, we denote $F_n:=(-)_n F$. We use the same notation for cosimplicial objects.

\noindent In the rest of this section we will direct our attention to two important examples of simplicial and cosimplicial objects. Where we study simplicial sets (a simplicial object in the category $\Sets$), and their topological realisation, induced by $\Delta$ a cosimplicial object of the category $\Top$.
we will recall the definition, basic properties and example in the following section.

\section{Simplicial Sets and Topological Realisation}
We saw in \ref{TriangulatedSpaces} that simplices plays the role of the blocks of the topological intuition "realisation" of the gluing data. We recall that the below defined functor $\Delta$ plays the same role, and gives  rise to the topological intuition about simplicial sets. Therefore, we start by recalling the definition of $\Delta$.\\

\noindent Consider the functor:
$$
\Delta:\Simp\rightarrow \Top
$$
given by $\Delta([n])=\Delta_n$, the n$^{th}$-simplex, and for every order preserving map $\mu:[m]\rightarrow [n]$, $\Delta(\mu)$ is defined by:
$$
\begin{array}{lcccl}
\Delta({\mu}):&\Delta_m&\rightarrow&\Delta_n\\
&u&\mapsto&t,&t_i=
\displaystyle\sum_{j\in {\mu^{-1}(i)}}u_j.
\end{array}
$$
We denote $\Delta_{\mu}:=\Delta(\mu)$. $\Delta$ is a functor that $\Delta_{id_{[n]}}=id_{\Delta_n}$, and for any composable pair of order preserving maps $[l]\stackrel{\nu}{\rightarrow}[m]\stackrel{\mu}{\rightarrow}[n]$, and every $v\in \Delta_l$, let $u=\Delta_{\nu}(v)$, $t=\delta_{\mu}(u)$, and $t'=\Delta_{\mu\nu}(v)$, then:\\
\[t_i=\displaystyle\sum_{j\in {\mu^{-1}(i)}}u_j\text{, and }u_j=\displaystyle\sum_{k\in {\nu^{-1}(j)}}v_k\text{, hence
}t_i=\displaystyle\sum_{j\in {\mu^{-1}(i)}}\displaystyle\sum_{k\in {\nu^{-1}(j)}}v_k=\displaystyle\sum_{k\in {(\mu\nu)^{-1}(i)}}v_k=t'_i\]
Hence,$\Delta_{\mu}\Delta_{\nu}=\Delta_{\mu\nu}$.
\noindent For every such morphism $\mu$, we understand $\Delta_{\mu}$ as an identification of the $m$-dimensional simplex with a face of the $n$-dimensional simplex, namely $\Imm \Delta_{\mu}$. Notice that $\Delta_{\mu}$ also maps the vertices $v_j$ to $v_{\mu(j)}$, on the contrary to the case of triangulated spaces, this mapping is not necessary injective. That, when $\mu$ is not strictly increasing the dimension of the face is less than the dimension of $\Delta_m$.\\
\begin{definition}[Simplicial Sets]
A simplicial set $X$ is defined to be simplicial object of $\Sets$:
$$X:\Simp^{op}\rightarrow\Sets$$
\end{definition}
\noindent One can think of simplicial sets as a graded set with maps between them, generated by $d_i^n$ and $s_i^n$ that satisfy \ref{SimpObRel}. We denote a generic element of $X_n$ by $x_n$. And we call morphisms of simplicial sets, simplicial \tcr{morphisms}. \\

\noindent We define the topological realisation of $X$ to be the topological space:
$$\mid X \mid=\left(\displaystyle\bigsqcup_{n=0}^{\infty}(\Delta_n\times X_{n,Dis})\right)/R$$
where $R$ is the smallest equivalence relation that identify each $(u,x_m)\in\Delta_m\times X_{m,Disc}$ and $(t,x_n)\in\Delta_n\times X_{n,Disc}$, such that there is a morphism $\mu:[m]\rightarrow [n]$ in $\Simp$ such that
\begin{equation}\label{eq_RealisationEq}
x_m=X_\mu(x_n),\ and \  t=\Delta_{\mu} (u).
\end{equation}
$\mid X \mid$ is topologies the same way as in \eqref{eq_TriRealisationTop}. Throughout these notes equivalence classes referred to by square brackets, so elements of $\mid X \mid$ are denoted $[(t,x_n)]$ where $t\in \Delta_n$ and $x_n\in X_n$.\\

\noindent The topological realisation allows us to think of elements of $X_n$ as $n\!-\!$simplices, glued using the above equivalence relation. Therefore, we call the elements of $X_n$ the $n$-simplices of $X$, in particular we call elements of $X_0$ the vertices of $X$. We drop the notation of the subscript $Dis$, and we use $X_n$ to refer to both the discrete topological space and its underlying set, when the it is not ambiguous.

\subsection{Degeneracy}
\noindent Let $X$ be a simplicial set, $x_{n+1}\in X_{n+1}$ is called degenerate simplex if there is $n\geq i\geq 0$, and $x_n\in X_n$ such that $x_{n+1}=s_i^n (x_n)$. That, in the realisation $\mid X \mid$, $(t,x_{n+1})$ is identified with $(s_n^i(t),x_n)$ for every $t\in \Delta_{n+1}$, so $x_{n+1}$ does not encode additional data more than these already given by $x_n$ and the degeneration. Notice that $0\!-\!$simplices of $X$ are all non-degenerate. The below lemma gives an alternative definition of degenerate simplices.

\begin{lemma}\label{Lem_Degeneration}
Let $X$ be a simplicial set, $x_m\in X_m$, then the following statements are equivalent:
\begin{enumerate}
\item $x_m$ is a degenerate simplex of $X$.
\item There exits $n<m, x_n\in X_n$ and a surjection $\mu:[m]\rightarrow [n]$, such that $x_m=X_{\mu}(x_n)$.
\item There exits $n<m, x_n\in X_n$ and a map $\mu:[m]\rightarrow [n]$, such that $x_m=X_{\mu}(x_n)$.
\end{enumerate}
\end{lemma}
\begin{proof}
\item[{$1\Rightarrow 2$, and $2\Rightarrow 3$}] are straightforward.
\item[{$3\Rightarrow 1$}] Assume there exits $n<m, x_n\in X_n$ and a map $\mu:[m]\rightarrow [n]$, such that $x_m=X_{\mu}(x_n)$. Then, using lemma \ref{SimpRelFactorisation}, $\mu$ factorise as:
$$\mu=\partial_n^{i_1}\partial_{n-1}^{i_2}...\partial^{i_s}_{n-s+1}\sigma^{j_t}_{m-t}\sigma^{j_{t-1}}_{m-t+1}...\sigma^{j_1}_{m-1}$$
If $t\leq 0$, then $n \geq m$, hence $t>0$, i.e. $\mu$ can be writen $\mu=\nu\sigma_{m-1}^i$, for some $0\leq i\leq m-1$, and $\nu:[m-1]\rightarrow [n]$. Then, $x_m=X_{\mu}(x_n)=s_i^{m-1}(X_{\nu}(x_n))$. Since, $X_{\nu}(x_n)\in X_{m-1}$, then $x_m$ is degenerate.
\end{proof}

\begin{lemma}[Eilenberg-Zilber Lemma]
Let $X$ be a simplicial set, for every $x_m\in X_m$ a $m\!-\!$simplex of $X$, there exists a unique surjection $\mu:[m]\rightarrow [n]$, and a unique non-degenerate $n\!-\!$simplex of $X$, $x_n\in X_n$, such that $x_m=X_{\mu}(x_n)$.
\tcr{cite}
\end{lemma}
\begin{proof}
\tcb{type}
\end{proof}
\noindent The above lemma says that a simplex of $X$ is either non-degenerate, or the "surjective"-image of unique non-degenerate given in the above canonical way. Therefore, in some occasions, it will be cnvennto direct our attention to the non-degenerate simplices of $X$, we denote the set of non-degenerate $n\!-\!$simplices by $X_{(n)}$.

\subsection{Generator of Simplicial Sets}
\tcr{find citation}
Let $X$ be a simplicial set, $S$ a graded subset set of $X$, i.e. $S_n\subset X_n, \forall n\in \ZZ_{\geq 0}$, not necessary a simplicial subset. For $[p]\in \Simp$, and $\theta:[q]\rightarrow [p]$ in $\Simp$, we define
$$
\begin{array}{l}
<S>_{X,p}=\{x_p\in X_p|\exists m \in \ZZ_{\geq 0} , \exists \beta:[p]\rightarrow [m]\text{ order-preserving }, \exists x_m\in S_m\text{, such that }x_p=X_{\beta}(x_m)\}\subset X_p.
\\
<S>_{X,\theta}(x_p)=X_{\theta}(x_p).
\end{array}
$$

\noindent $<S>_{X,\theta}$ is well defined that for $x_p\in <S>_{X,p}$, $\exists m \in \ZZ_{\geq 0} , \exists \beta:[p]\rightarrow [m], \exists x_m\in S_m$, such that $x_p=X_{\beta}(x_m)$, hence
$$
<S>_{X,\theta}(x_p)=X_{\theta}(x_p)=X_{\theta}(X_{\beta}(x_m))=X_{\beta\theta}(x_m)
$$
i.e. $\exists m \geq 0 , \exists \beta\theta:[q]\rightarrow [m]$ order-preserving, $\exists x_m\in S_m$, such that $<S>_{X,\theta}(x_p)=X_{\beta\theta}(x_m)$, hence $<S>_{X,\theta}(x_p)\in <S>_{X,q}$.\\ 

\noindent Since $X$ is simplicial set, then $<S>_X$, defined on objects and morphisms above, is so. Moreover, one can see readily that the above inclusions induce an inclusion of simplicial sets $<S>_X\subset X$. $<S>_X$ is  the smallest simplicial subset of $X$ that contains $S$ as a graded subset. 

\begin{definition}
We call $<S>_X$ defined above the simplicial subset of $X$ generated by $S$,
and we say that $X$ is generated by $S$ iff $X=<S>_X$, then we also say $S$ generates $X$, $S$ is a "generator" of $X$, and graded elements of $S$ are the "generators" of $X$.\\
We call a generator $S$ of $X$ a minimal generator if any graded subset $S'$ of $S$, that generates $X$, coincide with $S$.
\end{definition}

\begin{lemma}
Let $X$ be a simplicial set, then $X$ has a unique minimal generator.
\end{lemma}
\begin{proof}
Let $\bcS$ be the set of "generator"s of $X$, $\bcS\neq\emptyset$, that $<X>_X=X$, i.e. $X\in \bcS$. $\bcS$ is partially order by graded inclusion. Let $\bcT$ be a chain of totally ordered subset of $\bcS$, and let $S$ be the graded set defined by
$$
S_n=\displaystyle\bigcap_{T\in \bcT}T_n
$$
$S$ is a graded subset of $X$, \tcr{we need to show that $S$ generates $X$. If we do}, then by Zorn's Lemma, $\bcS$ has a minimal element, i.e. $X$ has a minimal generator.\\
\tcr{proof uniqueness, expressing identity as a composition of two order-preserving maps}.
\end{proof}

\noindent In all consider examples,  $S_n=\emptyset$ for all but finite many $n\in \ZZ_{\geq 0}$, In most cases $S_n\neq \emptyset$ only for one value of $n$.\\
\begin{remark}
Avoid the confusion between a generator of simplicial sets with the the generators of morphisms in the simplicial category.
\end{remark}

\begin{definition}
Let $X$ be a simplicial set, $x_n\in X_n,$ and $x_m\in X_m$, we define a simplicial relation of $x_n,$ and $x_m$ to be the triple $(p,\alpha,\beta)$ where $p\in \ZZ_{\geq}, \alpha:[p]\rightarrow [n],$ and $\beta:[p]\rightarrow [m]$ in $\Simp$, such that $X_{\alpha}(x_n)=X_{\beta}(x_m)\in X_p$. We say that there is a simplicial relation between $x_n,$ and $x_m$ in $X$, if such triple exists.\\
We say that a simplicial relation $(p,\alpha,\beta)$ factorises through a simplicial relation $(p',\alpha',\beta')$ between $x_n,$ and $x_m$, if there exists $i:[p]\rightarrow[p']$ in $\Simp$ such that the following diagram commutes:
$$
\xymatrix{
&[p]\ar[ddr]^{\beta}\ar[ddl]_{\alpha}\ar[d]|i&\\
&[p']\ar[dr]_{\beta'}\ar[dl]^{\alpha'}&\\
[n]&&[m]
}
$$
We say that a simplicial relation between $x_n,$ and $x_m$ is minimal if it does not factorise through different simplicial relation. Notice that when $(p,\alpha,\beta)$ factorise through $(p',\alpha',\beta')$, then:
$X_{\alpha}(x_n)=X_{\beta}(x_m)$ is a redundant simplicial relation that it can be deduced from $X_{\alpha'}(x_n)=X_{\beta'}(x_m)$.\\

Moreover, let $S$ be a graded subset of $X$, we define the set minimal simplicial relations on $S$ in $X$ to be the sets of all minimal simplicial relation between pairs of elements of $S$.
\end{definition}

\begin{lemma}
Let $X$ be a simplicial set, $x_n\in X_n$, and $x_m\in X_m$, then there is a unique minimal simplicial relation between $x_n$, and $x_m$.
\end{lemma}
\begin{proof}
\tcr{If not true, give a counter example}.
\end{proof}

\begin{lemma}
Let $X$ be a simplicial set, \tcr{$S$ is the generator of $X$}, then there is a unique set of minimal simplicial relations on $S$ in $X$.
\end{lemma}
\begin{proof}

\end{proof}
\tcb{te set of minimal simplicial relations can be the empty set}.
% prove teh legitimity of using the term minimal.


\tcr{Is there always a minimal (or minimum) generators}.
\tcr{link to generator in a category}


\begin{lemma}\label{Simp:Sets:MorphismsByGenerators}
Let $X$, and $Y$ be simplicial sets, $X$ generated by $S$. Then, giving a morphism of simplicial sets $f:X\rightarrow Y$ is equivalent to giving a graded map $f':S\rightarrow Y$, that respects the set of minimal simplicial relations on $S$ in $X$. Moreover, $f'$ is the restriction of $f$ to $S$.
\end{lemma}
\begin{proof}
It is clear that if $f:X\rightarrow Y$ is a morphism of simplicial sets, i.e. a natural transformation, then it is a graded map respects the set of minimal relations on $S$, and its restriction is so. Then, $f'=f|_S$.\\

\noindent On the other hand, assume that there is a graded map $f':S\rightarrow Y$, that respects the set of minimal simplicial relations on $S$ in $X$. Since $X$ is generated by $S$, then $\forall n\in \ZZ_{\geq 0},\forall x_p\in X_p, \exists n\in \ZZ_{\geq 0}, \exists \alpha:[p]\rightarrow [n], \exists x_n\in S_n$, such that $x_p=X_{\alpha}(x_n)$. Then we define:
$$
f_p(x_p)=Y_{\alpha}(f'_n(x_n))
$$
\noindent $f_p$ is well-defined. That, if there exist $m\in \ZZ_{\geq 0},  \exists \beta:[p]\rightarrow [m], x_m\in S_m$, such that $x_p=X_{\beta}(x_m)$. Then we have the simplicial relation between $x_n$ and $x_m$ given by $X_{\alpha}(x_n)=X_{\beta}(x_m)$. Since $f'$ respects minimal simplicial relations on $S$ in $X$, \tcb{then it respects all simplicial relations on $S$ in $X$}. Hence, $Y_{\alpha}(f'_n(x_n))=f'(X_{\alpha}(x_n))=f'(X_{\beta}(x_m))=Y_{\beta}(f'_m(x_m))$.

Then we have the graded map $f:X\rightarrow Y$, defined object-wise above. One sees readily that $f'=f|_S$. We need to show that $f$ is a morphism of simplicial sets, i.e. the following diagram commutes for every $\theta:[p]\rightarrow [q]$ 
$$
\xymatrix{X_p\ar[r]^{X_{\theta}}\ar[d]_{f_p}&X_q\ar[d]^{f_q}\\
Y_p\ar[r]_{Y_{\theta}}&Y_q
}
$$
\noindent
that 
$$\begin{array}{ll}
f_q(X_{\theta}(x_p))&=f_q(X_{\theta}(X_{\alpha}(x_n)))=f_q(X_{\alpha\theta}(x_n))=Y_{\alpha\theta}(f'(x_n))=Y_{\theta}(Y_{\alpha}(f'(x_n)))\\
&=Y_{\theta}(f_p(X_{\alpha}(x_n)))=Y_{\theta}(f_p(x_p)).
\end{array}
$$
Hence, $f$ is the desired morphism of simplicial sets.
\end{proof}
\begin{corollary}
Equality of morphisms of simplicial sets is determined on generators.
\end{corollary}

\begin{remark}
The same concept and results can be generalised for any functor $F:\bcC\rightarrow \SSets$, not necessary simplicial sets.
\end{remark}


\subsection{Topological Realisation and Singular Functor}
\begin{lemma}
Let $f:X\rightarrow Y$ be a morphism of simplicial sets, then $f$ induces functorially a continuous map between the topological realisations $\mid f\mid:\mid X \mid\rightarrow \mid Y\mid$, given by:
$$\mid f\mid \left([(t,x_n)]\right)=[(t,f_n(x_n))]$$
\end{lemma}
\begin{proof}
$\mid f\mid$ is well-defined that $\forall [(t,x_n)],[(u,x_m)]\in \mid X \mid$ such that $[(t,x_n)]=[(u,x_m)]$, then there exist an order-preserving map $\mu:[m]\rightarrow [n]$, or $\mu':[n]\rightarrow [m]$ that satisfies \ref{RealisationEq}. Without loose of generality, up to renaming, let $\mu:[m]\rightarrow [n]$ such that
$$
x_m=X_\mu(x_n),\ and \  t=\Delta_{\mu} (u),
$$
using \eqref{eq_SimpObMor} we see that $f_m(x_m)=f_m(X_\mu(x_n))=Y_{\mu}(f_n(x_n))$, hence $[(t,f_n(x_n))]=[(u,f_m(x_m))]$, i.e. $\mid f\mid \left([(t,x_n)]\right)=\mid f\mid \left([(u,x_m)]\right)$. \\

\noindent Consider the commutative diagram:
$$
\xymatrix{
\displaystyle\bigsqcup_{n=0}^{\infty}(\Delta_n\times X_n)\ar[rrr]^{\pi_X}\ar[d]_{f'}&&&\mid X \mid\ar[d]^{\mid f\mid }\\
\displaystyle\bigsqcup_{n=0}^{\infty}(\Delta_n\times Y_n)\ar[rrr]_{\pi_Y}&&&\mid Y\mid\\
}
$$
where $f'=\displaystyle\bigsqcup_{n=0}^{\infty}(id{\Delta_n}\times f_n)$, let $V\subset \mid Y\mid$ open, $(\mid f\mid \pi_X)^{-1}(V)=(\pi_Y f')^{-1}(V)$. $\pi_Y$ is continuous  by definition, and one can readily see that $f'$ is continuous, then $(\mid f\mid \pi_X)^{-1}(V)$ is open, hence $\mid f\mid ^{-1}(V)$ is open and $\mid f\mid $ is continuous.\\

One can readily see that $\mid id_X\mid=id_{\mid X \mid}$, and for any composable morphisms $X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z$, we have $\mid g f\mid=\mid g\mid \mid f\mid $. Hence, we have the functor
$$
|.|:\SSets\rightarrow \Top
$$
\end{proof}
\begin{definition}[Topological realisation]\label{TopologicalRealisation}
We call the above defined functor $|.|:\SSets\rightarrow \Top$, the topological realisation. This functor have a \tcr{right} adjoint, namely the singular homology functor. Both functor plays an essential role in the study of simplicial sets, and this adjuction induces an equivalence between the category of \tcr{Kan complexe}s and the category of \tcr{CW-complexes}.
\end{definition}
\tcr{Non-degenerate simplices}
\tcr{Give examples of different realisations rather than $\Delta$.}
\tcr{Kan extension, give example of a simplicial sets that does not satisfy Kan extension.}
\tcr{link with generators, it is just the simplices of the generators glued}..



\noindent Let $X$ be a topological space, $n\in \ZZ_{\geq 0}$, we define:
$$S(X)_{n}:=Hom_{\Top}(\Delta_n,X) =\{f_n:\Delta_n\rightarrow X|f_n\text{ is continuous}\},$$
and for every morphism $\mu:[m]\rightarrow [n]$ in $\Simp$, we define:
$$S(X)_{\mu}(f_n)=f_n\mu, f_n\in S(X)_{n}.$$
\noindent We can see readily that $S(X)_{id_{[n]}}=id_{S(X)_{n}}$, and $S(X)_{\mu\nu}=S(X)_{\nu} S(X)_{\mu}$, for composable morphisms $[l]\stackrel{\nu}{\rightarrow}[m]\stackrel{\mu}{\rightarrow}[n]$.
. Then, $$S(X):\Simp^{op}\rightarrow\Sets,$$ given on objects and morphisms above, forms a simplicial set, called the singular simplicial set of $X$. The $n$-simplices of $S(X)$ are called singular $n$-simplices of $X$.\\

For any continuous map of topological spaces $f:X\rightarrow Y$ we define
\[
\begin{array}{lclc}
S(f)_n:	&S(X)_n&\longrightarrow &S(Y)_n\\
		&f_n&\longmapsto&f f_n.
\end{array}
\]
The collection of $S(f)_n$ defines a morphisms of simplicial sets $S(F):S(X)\rightarrow S(Y)$, with componenets $S(f)_n$, that the following diagram commutes for every morphism $\mu:[m]\rightarrow [n]$ in $\Simp$:
\[
\xymatrix{S(X)_n\ar[r]^{S(f)_n}\ar[d]_{S(X)_{\mu}}&S(Y)_n\ar[d]^{S(Y)_{\mu}}\\
S(X)_m\ar[r]_{S(f)_m}&S(Y)_m
}.
\]
One can see easily that $S(id_X)=id_{S(X)}$, and that $S(g  f)=S(g) S(f)$ for composable contiuous maps $X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z$. Hence, we have the functor:
\[
S:\Top\rightarrow \SSets
\]
called the singular functor.
\noindent \tcr{Why is it called singular, the importance of its homology, mention it here as a motivation, and also as a motivation for Kan complexes, the adjunction.}

\begin{lemma}
The realisation functor is left adjoint to the singular functor, i.e. there is a natural isomorphism:
\[
\phi:\hom_{\Top}(\mid - \mid, -)\rightarrow \hom_{\SSets}(-,S(-))
\]
\end{lemma}
\begin{proof}
\tcr{....}
\end{proof}



\subsection{Generators and Topological realisations}
\tcb{The topological realisation is the generator simplices glued together by the minimal relations, the rest is just technical redundant information.}

\begin{lemma}
Let $X$ be a simplicial set, generated by $S$, then
$$\mid X \mid=\left(\displaystyle\bigsqcup_{n=0}^{\infty}(\Delta_n\times S_{n,Dis})\right)/R$$
Where, $R$ is the smallest equivalence relation generated by the set of minimal simplicial relations of $S$ in $X$.
\end{lemma}

\subsection{{\tcr{Morphisms and Operations of Simplicial Sets}}}
\noindent Let $X$ be a simplicial set, if there exists integer $n_0$ such that $X_{n_0}=\emptyset$, then $X_n=\emptyset \forall n\in \ZZ_{\geq 0}$, we call this simplicial set the empty simplicial set and we denote it by $\emptyset$. Also, we have the simplicial set $X$ given by $X=\ast,\forall n\in \ZZ_{\geq 0}$, where $\ast$ is a singleton, with the evident maps. We denote this simplicial set $\ast$. One can readily see that $\emptyset$, and $\ast$ are the initial and terminal objects of $\SSets$.

\noindent Natural transformations are monomorphisms, eipmorphisms, or isomorphisms if they are monomorphisms, eipmorphisms, or isomorphisms object-wise, respectively. Since, monomorphisms in $\Sets$ are just injections, we call a simplicial monomorphism $X\hookrightarrow Y$ a simplicial injection, and we denote it $X\subset Y$. Note that simplicial inclusion $X\subset Y$ is given by inclusions $X_n\subset Y_n, \forall n\in \ZZ_{\geq 0}$ that makes the  following diagram commute for every $\mu:[m]\rightarrow [n]$ in $\Simp$
\[
\xymatrix{
X_n\ar[d]_{X_{\mu}}\ar@{^(->}[rr]^{i_n}&&Y_n\ar[d]^{Y_{\mu}}\\
X_m\ar@{^(->}[rr]_{i_m}&&Y_n
}
\]
where $i_n$'s are the inclusion maps. We say that $X$ is a sub-simplicial set of $Y$.

\noindent Notice that not every such family of inclusions makes the above diagram commute. For example, let $X=\ast,\ast\neq Y\neq \emptyset$, then there exist $m\in \ZZ_{\neq 0}$ such that $\# Y_m>1$. Let $\mu:[m]\rightarrow [n]$ be in $\Simp$ for some $n\in \ZZ_{\geq 0}$, $y_n\in Y_n$, $y_m:=Y_{\mu}(y_n)$, then choose $y'_m\in Y_m$ different from $y_m$, and consider inclusions $i_n:\ast\rightarrow Y_n$, such that $\{y_n\}=\Imm i_n$, and $\{y'_m\}=\Imm i_m$, then the above diagram does not commute and the collection of maps $i_n$ does not define a morphism of simplicial sets.

\noindent We define operations on simplicial sets object-wise. Let $X$ and $Y$ be simplicial sets, then we define:
$$
(X\bigcap Y)_n:=X_n\bigcap Y_n\ \ ,\ \ (X\bigcup Y)_n=X_n\bigcup Y_n\ \ ,\ \ (X\setminus Y)_n=X_n\setminus Y_n
$$
for every $n\in \ZZ_{\geq 0}$. One can \tcr{easily} see that the result are well-defined simplicial sets, and that there exist injections $X\bigcap Y\subset X, X\subset X\bigcup Y$, and $X\setminus Y \subset X$.




\section{Classical Examples of Simplicial Sets}
Hereby we review several examples of simplicial sets. Some of these examples
plays an important role in the study of the canoincal model structure on the
category of simplicial sets, in \ref{SS_ModelStructure}, namely $n\!-\!$simplex $\Delta[n]$, skeleton, the boundary $\partial\Delta[n]$, and the $r\!-\!$horn $\Lambda^r[n]$. Also, we recall some other classical examples, interesting for their own.\\


\noindent We might not mention in this section that certain morphism between objects of $\Simp$ are order-preserving, to avoid long description, but it is always assumed to be the case, and it is essential in the following examples:

\subsection{{$n\!-\!$Skeleton and the Dimension of a Simplicial set}}
\noindent Let $X$ be a simplicial set, and $n\in \ZZ_{\geq 0}$. For $[p]\in \Simp$, and $\theta:[q]\rightarrow [p]$ in $\Simp$, we define
$$
\begin{array}{l}
(sk^nX)_p=\{x_p\in X_p|\exists m \leq n , \exists \beta:[p]\rightarrow [m]\text{ order-preserving }, \exists x_m\in X_m\text{, such that }x_p=X_{\beta}(x_m)\}\\
\ \ \ \subset X_p.
\\
(sk^nX)_{\theta}(x_p)=X_{\theta}(x_p).
\end{array}
$$
$(sk^nX)_{\theta}$ is well defined that for $x_p\in (sk^nX)_p$, $\exists m \leq n , \exists \beta:[p]\rightarrow [m], \exists x_m\in X_m$, such that $x_p=X_{\beta}(x_m)$, hence
$$
(sk^nX)_{\theta}(x_p)=X_{\theta}(x_p)=X_{\theta}(X_{\beta}(x_m))=X_{\beta\theta}(x_m)
$$
i.e. $\exists m \leq n , \exists \beta\theta:[q]\rightarrow [m]$ order-preserving, $\exists x_m\in X_m$, such that $(sk^nX)_{\theta}(x_p)=X_{\beta\theta}(x_m)$, hence $(sk^nX)_{\theta}(x_p)\in (sk^nX)_q$.

\noindent Since $X$ is simplicial set, then $sk^nX$, define on objects and morphisms above, is so. It is called the $n\!-\!$skeleton of $X$. Moreover, one can see readily that the above inclusions induce an inclusion of simplicial sets $sk^n X\subset X$.\\

\noindent Notice that $\forall p\leq n, \forall x_p\in X_p$, choosing $m=p$, and $\beta=id_{[p]}$, we see that $x_p\in (sk^nX)_p$. Hence, $(sk^nX)_p=X_p$.
\noindent Whereas, for $p>n, \forall x_p\in (sk^nX)_p$, by lemma \ref{Lem_Degeneration}, we see that $x_p$ is a degenerate simplex of $X$.

\noindent Hence, the simplices of $sk^nX$ are the $p\!-\!$simplices of $X$, with $p\leq n$, and their degenerations.

\begin{definition}[Dimension of Simplicial Sets]
\textup{Let $X$ be a simplicial set, $n\in\ZZ_{\geq 0}$ an integer such that $X=sk^nX\neq sk^{n-1}X$, then we say that $X$ has the dimension $n$.}
\end{definition}
\noindent When a simplicial set has dimension $n$, then all $p\!-\!$simplices are degenerate for $p>n$, and there exist a non-degenerate $n\!-\!$simplex.\\

\noindent \tcb{Notice that for $m<n$, we have the inclusion of simplicial sets $sk^m X\subset sk^n X$, and that $X=\displaystyle\colim_{\Simp}sk^{-}X=\displaystyle\bigcup_{n=0}^n sk^nX$.}\\

\noindent We can define $sk^{-1}X$ by extending the definition formally for $n=-1$, then $sk^{-1}X=\emptyset$.

\begin{lemma}
Let $X$ be a simplicial set of finite dimension, then the dimension of $X$ coincide with the topological dimension of its topological realisation $\mid X\mid $.
\end{lemma}
\tcr{Give example of infinite dimensional simplicial set., if any}

\subsection{$n\!-\!$Simplex $\Delta[n]$}
\noindent Let $n\in \ZZ_{\geq 0}$, we define the simplicial set $\Delta[n]$ to be the represented functor by $[n]$, i.e.
$$\Delta[n]=hom_{\Simp}(-,[n])$$
We will adopt the short notation $\Delta[n]_p:=\Delta[n]([p])$, and $\Delta[n]_{\theta}:=\Delta[n](\theta)$ for $[p]\in \Simp$, and $\theta:[q]\rightarrow [p]$ in $\Simp$. Then,\\
$$
\begin{array}{l}
\Delta[n]_p=\{\alpha:[p]\rightarrow [n]| \alpha\text{ is order-presrving}\}\cong\{(t_0,t_1,...,t_p))|n\geq t_p\geq ...\geq t_1\geq t_0\geq 0\}.
\\
\Delta[n]_{\theta}(\alpha)=\alpha\theta \in \Delta[n]_q \text{ for }\alpha \in \Delta[n]_p.
\end{array}
$$

\noindent Particularly, we have $\Delta[0]=\ast$ The above mention bijection shows that $\#\Delta[n]_p=\left(\begin{array}{c} n+p+1\\p+1\end{array}\right)$.\\

\noindent one can easily see that $id_{[n]}\in \Delta[n]_n$ is \tcr{the} generator of $\Delta[n]$, with an empty set of minimal simplicial relations. Then, by lemma \ref{Simp:Sets:MorphismsByGenerators} we see that morphisms of simplicial sets $f:\Delta[n]\rightarrow X$, are determined by a choice of $x_n\in X_n$ to be $x_n=f(id_{[n]})$\\
\begin{lemma}
\textup{Let $X$ be a simplicial set, then there is a canonical natural isomorphism
$$
\iota:X_-\stackrel{\cong}{\longrightarrow}hom_{\SSets}(\Delta[-],X)
$$}
\end{lemma}
\begin{proof}[Proof. 1]
This is a direct result of the Yoneda's lemma \ref{YonedasLemma}.
\end{proof}
\begin{proof}[Proof. 2]
$\forall n\in  \ZZ_{\geq 0}$ $\Delta[n]$ is generated by $id_{[n]}$, and the set of empty simplicial relations, then by lemma \ref{Simp:Sets:MorphismsByGenerators}, morphisms of simplicial sets $\Delta[n]\rightarrow X$ are in one-to-one correspondence with $n\!-\!$simplices of $X$, i.e.there is a bijection $$\iota_n:X_n\stackrel{\cong}{\longrightarrow}hom_{\SSets}(\Delta[n],X)$$
such that $(\iota_n(x_n))_n(id_{[n]})=x_n,\forall x_n\in X_n$. \\

\noindent Notice that $\forall \mu:[m]\rightarrow [n]$ in $\Simp$ the following diagram commutes
\[
\xymatrix{
X_n\ar[d]_{X_\mu}\ar[rr]^{\iota_n}&&hom_{\SSets}(\Delta[n],X)\ar[d]^{(\Delta[\mu])^{\ast}}\\
X_m\ar[rr]_{\iota_m}&&hom_{\SSets}(\Delta[m],X)
}.
\]
That lemma \ref{Simp:Sets:MorphismsByGenerators}, implies that morphisms of simplicial sets $\Delta[m]\rightarrow X$ coincide iff they coincide on the generator $id_{[m]}$ of $\Delta[m]$, and $\forall x_n\in X_n$, we have:
$$
\begin{array}{ll}
\left((\Delta[\mu])^{\ast}(\iota_n(x_n))\right)_m(id_{[m]})&=(\iota_n(x_n)\Delta[\mu])_m(id_{[m]})=
(\iota_n(x_n))_m(\Delta[\mu]_m(id_{[m]}))=(\iota_n(x_n))_m(\mu)\\&=(\iota_n(x_n))_m(\Delta[n]_\mu(id_{[n]}))
=X_\mu((\iota_n(x_n))_n(id_{[n]}))=X_{\mu}(x_n)\\&=\left(\iota_m (X_\mu(x_n))\right)_m(id_{[m]}).
\end{array}
$$
The last equality is due to the fact that $X_{\mu}(x_n)\in X_m$. Then, $(\Delta[\mu])^{\ast}(\iota_n(x_n))=\iota_m (X_\mu(x_n)), \forall x_n \in X_n$. Hence, $(\Delta[\mu])^{\ast}\iota_n=\iota_m X_\mu$, and the above diagram commutes. Then, the family $\iota_n$'s gives rise to the desired natural transformation.
\end{proof}
\tcr{Is it natural in both argument, i.e. in $X$?}\\

\noindent Based on the above lemma, we can abuse notation and call morphisms of simplicial sets $\Delta[n]\rightarrow X$ the $n\!-\!$simplices of $X$, referring to the above bijection $\iota_n$.

\noindent $\forall n\in \ZZ_{\geq 0}$, we adopted the convention of using the grade of a simplex of simplicial set in the name of that simplex, like $x_n\in X_n$, hence the notation $\iota_n(x_n)$ becomes redundant. Therefore, we shorten notation, and adopt the convention of writing $\iota_{x}:=\iota_n(x), \forall x \in X_n \ZZ_{\geq 0}$, whenever the grade of $x$ is known, either appearing in the name like $x_n$, or can be calculated like $X_\mu(x_n)$, for $\mu:[m]\rightarrow [n]$ in $\Simp$. We refer to the $p^\text{th}\!-\!$component of $\iota_{x_n}$ by $\iota_{x_n,p}$. So, in particular, we have $\iota_{x_n,n}(id_{[n]})=x_n, \forall x_n \in X_n$.\\

\noindent \tcb{In particular}, morphisms of simplicial sets $\Delta[0]\rightarrow X$ are identified with vertices of $X$. 
\tcr{motivation for the following definition}
\begin{definition}[Fibers of morphism of simplicial sets]
Let $f:X\rightarrow Y$ be a morphism of simplicial sets, $y_0\in Y_0$ a vertex of $Y$, if there exist a cartesian square:
\[
\xymatrix{
X_{y_0}\ar[r]\ar[d]&X\ar[d]^f\\
\Delta[0]\ar[r]_{\iota_{y_0}}&Y}
\]
we call $X_{y_0}$ the fiber of $f$ at $y_0$.
\end{definition}

\noindent $id_{[n]}\in \Delta[n]_n$ is non-degenerate, that if we assume for the sake of contradiction that it is degenerate, then there exists $n'<n$, and an order-preserving map $\mu:[n]\rightarrow [n']$, and $\alpha \in \Delta[n]_{n'}$ such that $id_{[n]}=\Delta[n]_{\mu}(\alpha)=\alpha \mu$. $\mu$ is not an injection because $n'<n$, and that contradicts with the previous equality. Hence, $id_{[n]}$ is non-degenerate.\\

\noindent For every integer $p>n$, every $p\!-\!$simplex is degenerate. $\forall \alpha \in \Delta[n]_p$, $\alpha=id_{[n]}\alpha=\Delta[n]_{\alpha}(id_{[n]})$. Since $p>n$, then $\alpha$ is degenerate, by lemma \ref{Lem_Degeneration}.\\

\noindent Therefore, ${\Delta[n]}_{(n)}\neq \emptyset$, and ${\Delta[n]}_{(p)}=\emptyset$ for $p>n$. Hence $\dim\ \Delta[n]=n$.\\

\noindent The $n\!-\!$Simplex is very important simplicial set,we will see later that $\Delta[1]$ plays in simplicial homotopy theory the role of the unite interval in the topological homotopy theory. We recall some of the facts about it in the following lemmas:
\begin{lemma}\label{Lem:RealOFDelta[n]}
\textup{The topological realisation of the $n\!-\!$simplex $\Delta[n]$ is the standard $n\!-\!$simplex $\Delta_n$}.
\end{lemma}
\begin{proof}
\noindent Consider the maps:
\[
\begin{array}{cccc}
f:	&\Delta_n	&\longrightarrow&\mid \Delta[n] \mid\\
	&t			&\mapsto	&[(t,id_{[n]})]
\end{array}
\ \ \ \ \ \ 
\begin{array}{cccc}
g:	&\mid \Delta[n] \mid &\longrightarrow&\Delta_n\\
	&[(s,\alpha)]			&\mapsto	&\Delta_{\alpha}(s)
\end{array}
\]
\noindent One can see readily that $f$ is a continuous map. $g$ is well-defined that $\forall [(s,\alpha)],[(s',\alpha')]\in \mid \Delta[n] \mid$, such that $[(s,\alpha)]=[(s',\alpha')]$, then, without loose of generality and up-to renaming, $\exists [p],[p']\in \Simp$, and $\theta:[p']\rightarrow [p]$, such that $\alpha\in\Delta[n]_p$, $\alpha'\in\Delta[n]_{p'}$, and
$$s=\Delta_{\theta}(s')\text{, and }\alpha'=\Delta[n]_{\theta}(\alpha)=\alpha \theta $$
Hence, $$g([(s',\alpha')])=\Delta_{\alpha'}(s')=\Delta_{\alpha \theta}(s')=
\Delta_{\alpha} \left(\Delta_{\theta}(s')\right)=
\Delta_{\alpha}(s)=g([(s,\alpha)])$$
Also, $g$ is continuous, that projections and $\Delta_{\alpha}$'s are continuous.

\noindent In order to prove the lemma we need to see that $g$ is the inverse of $f$:\\
\noindent $\forall t\in \Delta,$ $$(g f)(t)=g([(t,id_{[n]})])=\Delta_{id_{[n]}}(t)=id_{\Delta_n}(t),$$hence $g circ f=id_{\Delta_n}$

\noindent On the other hand $\forall [(s,\alpha)]\in \mid \Delta[n] \mid, \exists [p]\in \Simp$ such that $\alpha\in \Delta[n]_p$ and $s\in \Delta_n$. Then, 
$$(f g)([(s,\alpha)])=f(\delta_{\alpha(s)})=[(\delta_{\alpha(s)},id_{[n]})].$$
Since $\alpha=\Delta[n]_{\alpha}(id_{[n]})$ , then 
$$(f g)([(s,\alpha)])=[(\delta_{\alpha(s)},id_{[n]})]=[(s,\alpha)]=id_{\mid \Delta[n] \mid}([(s,\alpha)]),$$ hence $f g=id_{\mid \Delta[n] \mid}$, and $\Delta_n$ is homeomorphic to $\mid \Delta[n] \mid$.
\end{proof}
\noindent The proof of above lemma shows that the $id_{[n]}\in \Delta[n]_n$ correspond to the the standard $n\!-\!$simplex $\Delta_n$, whereas all other simplices of $\Delta[n]$ are either faces of $id_{[n]}$, or degenerations of $id_{[n]}$ and its faces.
\begin{lemma}\label{Lem:SingOFDelta[n]}
\textup{The $n\!-\!$simplex $\Delta[n]$ is the singular simplicial set of the standard $n\!-\!$simplex $\Delta_n$.}
\end{lemma}
\begin{proof}
\tcb{type}
\end{proof}
\tcr{lemma 2.1 Jardine}\\\\



\noindent For any $\mu:[m]\rightarrow [n]$ in $\Simp$, $[p]\in \Simp$ consider the map $\Delta[\mu]_p:\Delta[m]_p\rightarrow\Delta[n]_p$ given by $\Delta[\mu]_p(\beta)=\mu\beta$ for $\beta\in \Delta[m]_p$. Then, once can see easily that the following diagram commutes for any $\theta:[q]\rightarrow [p]$:
$$
\xymatrix{\Delta[m]_p\ar[rrr]^{\Delta[\mu]_p}\ar[d]_{\Delta[m]_{\theta}}&&&\Delta[n]_p\ar[d]^{\Delta[n]_{\theta}}\\
\Delta[m]_q\ar[rrr]_{\Delta[\mu]_p}&&&\Delta[n]_q\\
}
$$
Hence, the set of maps $\Delta[\mu]_p$ defines a morphism of simplicial sets $\Delta[\mu]:\Delta[m]\rightarrow\Delta[n]$. One can readily see that the  this defines a functor $\Delta[-]:\Simp\rightarrow\SSets$. $\Delta[-]$ is faithful full that:
\noindent For $\mu,\mu':[m]\rightarrow [n]$ such that $\Delta[\mu]=\Delta[\mu']$, then in particular $\Delta[\mu]_m=\Delta[\mu']_m$, and 
$$
\mu=\Delta[\mu]_m(id_{[m]}=\Delta[\mu']_m(id_{[m]}=\mu'.
$$
\noindent On other hand, morphisms of simplicial sets $\Delta[m]\rightarrow\Delta[n]$ are in one to one correspondence to $m\!-\!$simplices of $\Delta[n]$, to be the image of $id_{[m]}$, i.e. they are in one to one correspondence to morphisms $\mu:[m]\rightarrow [n]$ in $\Simp$. Since, each such $\mu$ induces $\Delta[\mu]$, then each morphism of simplicial sets $\Delta[m]\rightarrow\Delta[n]$ is of the form $\Delta[\mu]$, for a $\mu$ in $\Simp$. I.e. $\Delta[-]$ is a full embedding of categories.

\noindent We tend to abuse notation and denote by $\Simp$ both the simplicial category and the full subcategory the image of $\Delta[-]$.

\begin{definition}[{The Simplex category of $X$, $(\Delta[-]\downarrow X)$}]
\noindent Let $X$ be  a simplicial set, we define the simplex category of $X$ to be the comma category $(\Delta[-]\downarrow X)$, with objects being the morphisms of simplicial sets of the form  $\Delta[n]\rightarrow X$, i.e. $\iota_{x_n}$ for all $x_n\in X_n, n\in \ZZ_{\geq 0}$, and morphisms being commutative diagrams:
$$
\xymatrix{
\Delta[m]\ar[rr]^f\ar[dr]_{\iota_{x_m}}&&\Delta[n]\ar[dl]^{\iota_{x_n}}\\
&X\ar@{}[u]|D&
}
$$
,induced by a unique morphism $\mu:[m]\rightarrow [n]$ in $\Simp$, as we have see above.\\
Then, we define the evident forgetful functor $U_X:(\Delta[-]\downarrow X) \rightarrow \SSets$, given by 
$$\begin{array}{rcl}
U_X(\iota_{x_n})&=&\Delta[n]\\
U_X(\,D\ )&=&f.
\end{array}
$$.
\end{definition}

\noindent The below lemma shows the significance of $\Delta[n]$, and indicates that all simplicial sets are just a colimit of diagrams of $\Delta[n]$'s.
\begin{lemma}
Let $X$ be a simplicial set, then $X=\displaystyle\colim_{(\Delta[-]\downarrow X)}U_X$.
\end{lemma}
\begin{proof}
Consider the constant functor $\hat{X}:(\Delta[-]\downarrow X) \rightarrow \SSets$ given by
$$
\hat{X}(\iota_{x_n})=X\ \text{, and }\ \hat{X}(D)=id_X.
$$
\noindent Define the family of morphisms of simplicial sets  $$\eta_{X,\iota_{x_n}}=\iota_{x_n}:\Delta[n]\rightarrow X.$$
Then, by the definition of $(\Delta[-]\downarrow X)$ the below diagram commutes for any $D:\iota_{x_m}\rightarrow \iota_{x_n}$ in $(\Delta[-]\downarrow X)$
$$
\xymatrix{
\Delta[m]=U_X(\iota_{x_m})\ar[rr]^{\eta_{X,\iota_{x_m}}=\iota_{x_m}}\ar[d]_{U_X(D)}&&\hat{X}(\iota_{x_m})=X\ar@{=}[d]\\
\Delta[n]=U_X(\iota_{x_n})\ar[rr]_{\eta_{X,\iota_{x_n}}=\iota_{x_n}}&&\hat{X}(\iota_{x_n})=X\\
}
$$
Hence, this family of morphisms gives rise to a natural transformation $\eta_X:U_X\stackrel{\cdot}{\rightarrow} \hat{X}$ given component-wise above.\\

\noindent In order to show that the desired colimit is $X$, we need to how that the pair $(X,\eta_X)$ is initial universal among such pairs. Let $Y$ be a simplicial set and $\eta_Y:U_X\stackrel{\cdot}{\rightarrow} \hat{Y}$ a natural transformation, where $\hat{Y}:(\Delta[-]\downarrow X) \rightarrow \SSets$ is the constant functor given by
$$
\hat{Y}(\iota_{x_n})=Y\ \text{, and }\ \hat{Y}(D)=id_Y.
$$
\noindent We need to show that $(Y,\eta_Y)$ factorise through $(X,\eta_X)$ uniquely. We can define the family of maps $\zeta_n:X_n\rightarrow Y_n$ given by $$\zeta_n(x_n)=\eta_{Y,\iota_{x_n},n}(id_{[n]}).$$
For any $\mu:[m]\rightarrow [n]$ in $\Simp$, the following diagram commutes:
$$
\xymatrix{
X_n\ar[rr]^{\zeta_n}\ar[d]_{X_{\mu}}&&Y_n\ar[d]^{Y_{\mu}}\\
X_m\ar[rr]_{\zeta_m}&&Y_m
}
$$
\noindent That $\forall x_n\in X_n$:
$$
Y_{\mu}(\zeta_n(x_n))=Y_{\mu}(\eta_{Y,\iota_{x_n},n}(id_{[n]}))=\eta_{Y,\iota_{x_n},m}(\Delta[n]_{\mu}(id_{[n]}))=\eta_{Y,\iota_{x_n},m}(id_{[n]}\mu)=\eta_{Y,\iota_{x_n},m}(\mu)
$$
The second equality is due to the fact that $\eta_{Y,\iota_{x_n}}$'s, the component of the natural transformation $\eta_Y$, are morphisms of simplicial sets, i.e. natural transformations. On the other hand:
\begin{align}\label{eq_Colim_2}
\zeta_m(X_\mu(x_n))=\eta_{Y,\iota_{X_\mu(x_n)},m}(id_{[m]})=\eta_{Y,\iota_{x_n}\Delta[\mu],m}(id_{[m]})\ \ \ ...\ \ \ ({\ast})
\end{align}

Te second equality is due to the fact that $\iota_n$ is a natural transformation (actually a natural isomorphism), i.e. $\iota_m X_{\mu}=(\Delta[\mu])^{\ast}\iota_n$, hence, $\iota_{X_\mu(x_n)}=\iota_{x_n}\Delta[\mu]$. We have the following commutative diagram in $(\Delta[-]\downarrow X)$:
$$
\xymatrix{
\Delta[m]\ar[rr]^{\Delta[\mu]}\ar[dr]_{\iota_{x_n}\Delta[\mu]}&&\Delta[n]\ar[dl]^{\iota_{x_n}}\\
&X\ar@{}[u]|D&
}
$$
Then, since $\eta_Y$ is a natural transformation, we have the commutative diagram:
$$
\xymatrix{
\Delta[m]=U_X(\iota_{x_n}\Delta[\mu])\ar[rr]^{\eta_{Y,\iota_{x_n}\Delta[\mu]}}\ar[d]_{\Delta[\mu]=U_X(D)}&&\hat{Y}(\iota_{x_n}\Delta[\mu])=Y\ar@{=}[d]\\
\Delta[n]=U_X(\iota_{x_n})\ar[rr]_{\eta_{Y,\iota_{x_n}}}&&\hat{Y}(\iota_{x_n})=Y\\
}
$$
Substituting in \ref{eq_Colim_2}, we find that:
$$
\zeta_m(X_\mu(x_n))=\eta_{Y,\iota_{x_n},m}(\Delta[\mu](id_{[m]}))=\eta_{Y,\iota_{x_n},m}(\mu)
$$
i.e. the family $\zeta_n$'s gives rise to a morphism of simplicial sets $\zeta:X\rightarrow Y$, and hence to the constant natural transformation $\hat{\zeta}:\hat{X}\stackrel{\cdot}{\rightarrow} \hat{Y}$, with $\hat{\zeta}_{\iota_{x_n}}=\zeta$.\\


\noindent We need to show that $\eta_Y=\hat{\zeta} \eta_X$, i.e. we need to show that $\eta_{Y,\iota_{x_n},m}=\zeta_{m} \eta_{X,\iota_{x_n},m}, \forall m\in \ZZ_{\geq 0},\forall x_n\in X_n, \forall n\in \ZZ_{\geq 0}$. Then, $\forall \mu\in \Delta[n]_m$, we have:
$$
\begin{array}{ll}
\zeta_{m}( \eta_{X,\iota_{x_n},m}(\mu))&=
\zeta_{m}(\iota_{x_n,m}(\mu))=\zeta_{m}(\iota_{x_n,m}(\Delta[n]_{\mu}(id_{[n]})))=
\zeta_{m}(X_{\mu}(\iota_{x_n,n}(id_{[n]})))\\&=\zeta_{m}(X_{\mu}(x_n))=Y_{\mu}(\zeta_n(x_n))=Y_{\mu}(\eta_{Y,\iota_{x_n},n}(id_{[n]}))=\eta_{Y,\iota_{x_n},m}(\Delta[n]_{\mu}(id_{[n]}))\\&=\eta_{Y,\iota_{x_n},m}(\mu)
\end{array}
$$
The above equalities follows from the commutativity of the diagram:

\[
\xymatrix @R=.75pc{
&&\Delta[n]_n\ar[lldddd]|{\eta_{X,\iota_{x_n},n}}\ar[dd]|{\Delta[n]_{\mu}}\ar[rrdddd]|{\eta_{Y,\iota_{x_n},n}}&&\\
\\
&&\Delta[n]_m\ar[lldddd]|{\eta_{X,\iota_{x_n},m}}\ar[rrdddd]|{\eta_{Y,\iota_{x_n},m}}&&\\
\\
X_n\ar[dd]_{X_{\mu}}\ar@{..>}[rrrr]^{\zeta_n}&&&&Y_n\ar[dd]^{Y_{\mu}}\\
\\
X_m\ar[rrrr]_{\zeta_m}&&&&Y_m
}
\]
\noindent Hence, $\eta_Y=\hat{\zeta} \eta_X$.\\

\noindent To show the uniqueness of $\hat{\zeta}$, let $\hat{\zeta}':\hat{X}\stackrel{\cdot}{\rightarrow}\hat{Y}$ be a natural transformation such that $\eta_Y=\hat{\zeta}'\eta_X$. Since $\hat{X},$ and $\hat{Y}$ are constant functors, then $\hat{\zeta}$ is a constant natural transformation. Let $\zeta':=\hat{\zeta}'_{\iota_{x_n}},\forall x_n\in X_n, \forall n\in \ZZ_{\geq 0}$. Then
$\zeta'_{m} \eta_{X,\iota_{x_n},m}=\eta_{Y,\iota_{x_n},m}, \forall m\in \ZZ_{\geq 0},\forall x_n\in X_n, \forall n\in \ZZ_{\geq 0}$
Then, $\forall n\in \ZZ_{\geq 0},\forall x_n\in X_n$, we have
$$
\zeta'_n(x_n)=\zeta'_n(\iota_{x_n,n}(id_{[n]}))\zeta'_n(\eta_{X,\iota_{x_n},n}(id_{[n]}))=\eta_{Y,\iota_{x_n},n}(id_{[n]})=\zeta_n(x_n)
$$
Hence $\zeta=\zeta'$, and $\hat{\zeta}=\hat{\zeta}'$. Therefore, $X=\displaystyle\colim_{(\Delta[-]\downarrow X)}U_X$.
\end{proof}



\subsection{Boundary $\partial\Delta[n]$}
\noindent Let $n\in \ZZ_{\geq 0}$, we define the boundary of the $n\!-\!$simplex $\Delta[n]$ to be the $(n-1)\!-\!$skeleton of $\Delta[n]$, and we denote it $\partial\Delta[n]$, i.e. 
$$\partial\Delta[n]=sk^{n-1}\Delta[n]$$
Then, for every $p\geq 0$ we have:
$$
\begin{array}{ll}
\partial\Delta[n]_p&=\{\alpha:[p]\rightarrow [n]| \exists m \leq n-1 , \exists \beta:[p]\rightarrow [m], \exists \mu:[m]\rightarrow [n]\text{, such that }\alpha=\mu\beta\}\\
&=\{\alpha:[p]\rightarrow[n]|\Imm \alpha\neq [n]  \}=\{\alpha:[p]\rightarrow[n]|\alpha\text{ is not surjective}\}.\\
\partial\Delta[n]_{\theta}(\alpha)&=\alpha\theta.
\end{array}
$$

\noindent one can easily see that $\partial\Delta[n]$ is the simplicial subset of $\Delta[n]$, generated by $\partial_n^i\in \Delta[n]_{n-1}$ for $0\leq i\leq n$ with is following set of minimal relations:
$$
\partial_{n}^j\partial_{n-1}^i=\partial_n^i\partial_{n-1}^{j-1}\ \ \text{, i.e.   }\ \ d^{n-1}_i(\partial_{n}^j)=d^{n-1}_{j-1}(\partial_{n}^i)\ \  \text{ for }i<j
$$
\noindent Notice that $\partial\Delta[0]=\emptyset$.
\begin{remark}
Avoid the confusion of thinking about the following relations as a simplicial relations between the generators $\partial_{n}^i$'s.
$$
\sigma_{n-1}^j \partial_n^i=\left\{
\begin{array}{ll}
\partial_{n-1}^i\sigma_{n-2}^{j-1} &\text{for } i<j\\
id_{[n-1]}&\text{for } i=j \text{ or } i=j+1\\
\partial_{n-1}^{i-1}\sigma_{n-2}^{j}&\text{for } i>j+1
\end{array}\right.
$$
They are rather relations between $\partial_{n-1}^i$'s, $\sigma_{n-1}^i$'s, and $id_{[n-1]}$ that their induced simplicial relations are:
$$
d^n_i(\sigma_{n-1}^j)=\left\{
\begin{array}{ll}
s^{n-2}_{j-1}( \partial_{n-1}^i)&\text{for } i<j\\
\Delta[n]_{id_{[n-1]}}(id_{[n-1]})&\text{for } i=j \text{ or } i=j+1\\
s^{n-2}_{j} (\partial_{n-1}^{i-1})&\text{for } i>j+1
\end{array}\right.
$$
\end{remark}

\noindent The generators $\partial\Delta[n]$ are all $(n-1)\!-\!$simplices, hence dim $\partial\Delta[n]=n-1$.\\

Recall that the boundary of the standard $n\!-\!$simplex $\partial\Delta_n$ is given by:
$$
\partial\Delta_n=\{t\in \Delta_n|\exists 0\leq i\leq n\text{, such that }t_i=0\}.
$$
\begin{lemma}
The topological realisation of $\partial\Delta[n]$ is the boundary of the standard $n\!-\!$simplex $\partial\Delta_n$.
\end{lemma}
\begin{proof}
Similar to the proof of lemma \ref{Lem:RealOFDelta[n]} using the well defined maps:
$$
\begin{array}{ccccc}
f:	&\partial\Delta_n	&\longrightarrow&\mid \partial\Delta[n] \mid&\\
	&t			&\mapsto	&[(t,\partial_n^i)]&\text{ where } t^i=o, \text{ for some }0\leq i\leq n
\end{array}
\ \ \ \ \ \ 
\begin{array}{cccc}
g:	&\mid \partial\Delta[n] \mid &\longrightarrow& \partial\Delta_n\\
	&[(s,\alpha)]			&\mapsto	&\Delta_{\alpha}(s)
\end{array}
$$
\end{proof}

\begin{lemma}
\textup{The boundary of $n\!-\!$simplex $\partial\Delta[n]$ is the singular simplicial set of the boundary standard $n\!-\!$simplex $\delta\Delta_n$.}
\end{lemma}

\noindent $\Delta[1]$ plays the role of the unit interval, we distinguish from its boundary $\partial\Delta[n]$, two $0\!-\!$simplices, $\partial_1^0$ corresponding to deleting the vertex $v_0$, so we call it $(1)$, and $\partial_1^1$ corresponding to deleting the vertex $v_1$, so we call it $(0)$. 

\noindent When we write $\partial\Delta[n]\hookrightarrow \Delta[n]$, we refer
to the canonical inclusion, unless mentioned otherwise. One can see readily that
$\forall n\in \ZZ_{\geq 0},\forall \mu:[n]\rightarrow [n]$, we have a morphism
of simplicial sets $i_\mu:\partial\Delta[n]\rightarrow \Delta[n]$ that sends
$\partial_n^i$ to $\mu\partial_n^i$ for $0\leq i\leq n$.
\begin{lemma}
$\forall n\in \ZZ_{\geq 0}$, let $f:\partial\Delta[n]\rightarrow \Delta[n]$ be a
morphism of simplial sets. Then, there exists $\mu:[n]\rightarrow [n]$ such that
$f=i_\mu$.
\end{lemma}
\begin{proof}
\tcr{type.}
\end{proof}
\subsection{$r\!-\!$horn $\Lambda^r[n]$}
\noindent Let $n\in \ZZ_{\geq 0}$, $0\leq r \leq n$, we define $\Lambda^r[n]$ to be the simplicial subset of $\Delta[n]$ generated by $\partial_n^i\in \Delta[n]_{n-1}$ for $0\leq i\neq r\leq n$ with is following set of minimal relations:
\begin{equation}\label{Eq:G&R:Horn}
d^{n-1}_i(\partial_{n}^j)=d^{n-1}_{j-1}(\partial_{n}^i)\ \  \text{ for }i<j\text{, and } i,j\neq r
\end{equation}
\begin{lemma}
Let $n\in \ZZ_{\geq 0}, 0\leq r\leq n$, then:
$\Lambda^r[n]_p=\{\alpha:[p]\rightarrow[n]|\alpha\text{ is order-preserving},[n]\neq\Imm \alpha\neq [n] \setminus \{r\} \}$
\end{lemma}
\begin{proof}

\end{proof}

\begin{lemma}
Let $n\in \ZZ_{\geq 0}, 0\leq r \leq n$, the topological realisation of $\Lambda^r[n]$ is $\Lambda^r\Delta_n$, i.e. the boundary of $\Delta_n$, with the face $r$, opposite to the vertex $r$, deleted.
\end{lemma}

\noindent When we write $\Lambda^r[n]\hookrightarrow \Delta[n]$, we refer to the canonical inclusion, unless mentioned otherwise.
\gray
%\subsection*{Nerve of Covering}
%\subsection*{The Simplicial Set Associated to a Triangulated Space}
\subsection*{Classifying Space of a Group}
\subsection*{Nondegenerate Simplices}
\tcb{Generalise to Grothendieck settings, using initial objects instead of empty sets.}
\black 



\section{Examples of Simplicial Objects}
\noindent Let $X$ be a simplicial set, $C:\Sets\rightarrow \Ab$ a functor. Then, $CX$ is a simplicial abelian group.\\
\noindent The following examples is of a particular interest that it would be used to define homology theory of simplicial sets. Let $X$ be a simplicial set, $A$ an abelian group, consider the functor $C_A:\Sets\rightarrow \Ab$, that sends each set to its free generated group, with coefficients in $A$. Then $C_AX$ is a simplicial abelian group.\\
\noindent The elements of $C_n(X,A):=C_AX([n])$ are finite formal sums of elements of $X_{(n)}$, with coefficients in $A$, let $c\in C_n(X,A)$, then $$c=\displaystyle\sum_{x\in X_{(n)}}a_x x,a_x\in A.$$\\

\noindent \tcr{We need also to consider $\Simp\rightarrow \bcC$, then we can also consider the functor $\Hom(-,A):\Ab^{op}\rightarrow \Ab$, then we have the simplicial abelian group $\Hom(C_AX,A)$. in order to introduce cohomology theory of simplicial sets.}
\section{Abstract Realisations}
\noindent When we examine topological realisation \ref{TopologicalRealisation}, we notice that the realisation makes sense in a more general settings. To see that, we notice first that the category of topological spaces $\Top$ are complete and cocomplete category, so are $\Top^{op}$ and $\Top\times\Top^{op}$. We have the forgetful functor
$$
U:\Top\rightarrow \Sets
$$			
and its left adjoint
$$
F:\Sets\rightarrow \Top
$$
that sends each set $S$ to the discrete topological space $S_{Dis}$. Then, notice that in $\mid X \mid$, $(u,y)\in\Delta_m\times X_{(m),Dis}$ and $(t,x)\in\Delta_n\times X_{(n),Dis}$ are identified when there exists a morphism $\nu:[m]\rightarrow [n]$ in $\Simp$, such that
$$y=X(\nu)(x),\ and \  t=\Delta_{\nu} (u)$$,
when we consider $X^{op}:\Simp\rightarrow\Top^{op}$, the above relations can be simplified to:
$$
\tcr{(t,x)=\left((\Delta,F^{op}(X^{op}))(\mu)\right)(u,y)}.
$$
Hence, we can see that the topological realisation of the simplicial set $X$ is given by
$$
\tcr{\mid X \mid=\displaystyle\Colim_{\Simp}\Delta\times F^{op}(X^{op})}
$$
Actually, we call it the discrete topological realisation, that it depends on $F$.\\

\tcb{\noindent Let $\bcD$ be a complete and cocomplete category, $F:\bcC\rightarrow \bcD$ a functor, $X$ a simplicial object in $\bcC$, and $R$ a cosimplicial object of $\bcD$, then we can define the $F\!-\!$realisation of $X$ with realisation "blocks" $R$ to be:
$$
\mid X \mid=\displaystyle\Colim_{\Simp}R\times F^{op}(X^{op})
$$}
\tcr{Study properties and give examples, and correct it in the case of topological realisation.}
\section{Complexes of a Simplicial Sets}
\gray
\subsection{Chain Complexes}
Let $C:\Simp^{op}\rightarrow \Ab$ be a simplicial abelian group. We define $d_n:C_{(n)}\rightarrow C_{(n-1)}$:
$$d_n=\displaystyle\sum_{i=0}^n(-1)^iC(\partial_n^i),$$ Which turns $$...\rightarrow C_{(n+1)}\stackrel{d_{n+1}}{\rightarrow} C_{(n)}\stackrel{d_{n}}{\rightarrow} C_{(n-1)}\rightarrow...$$ into a chain complex that, for $n\geq 1$,\\

\noindent${ \begin{array}{ll}
d_{n-1}d_n&=\left(\displaystyle\sum_{j=0}^{n-1}(-1)^jC(\partial_{n-1}^j)\right) \left(\displaystyle\sum_{i=0}^n(-1)^iC(\partial_n^i)\right)=\displaystyle\sum_{i=0}^n\displaystyle\sum_{j=0}^{n-1}(-1)^{i+j}C(\partial_n^i\partial_{n-1}^j)\\&=
\displaystyle\sum_{i=1}^n\displaystyle\sum_{j=0}^{i-1}(-1)^{i+j}C(\partial_n^i\partial_{n-1}^j)+\displaystyle\sum_{i=0}^{n-1}\displaystyle\sum_{j=i}^{n-1}(-1)^{i+j}C(\partial_n^i\partial_{n-1}^j)\\&=
\displaystyle\sum_{i=1}^n\displaystyle\sum_{j=0}^{i-1}(-1)^{i+j}C(\partial_n^j\partial_{n-1}^{i-1})+\displaystyle\sum_{i=0}^{n-1}\displaystyle\sum_{j=i}^{n-1}(-1)^{i+j}C(\partial_n^i\partial_{n-1}^j)\\&=
\displaystyle\sum_{k=0}^{n-1}\displaystyle\sum_{j=0}^{k}(-1)^{k+j+1}C(\partial_n^j\partial_{n-1}^k)+\displaystyle\sum_{i=0}^{n-1}\displaystyle\sum_{j=i}^{n-1}(-1)^{i+j}C(\partial_n^i\partial_{n-1}^j)\\&=
\displaystyle\sum_{j=0}^{n-1}  \displaystyle\sum_{k=j}^{n-1}  (-1)^{k+j+1}C(\partial_n^j\partial_{n-1}^k)+\displaystyle\sum_{i=0}^{n-1}\displaystyle\sum_{j=i}^{n-1}(-1)^{i+j}C(\partial_n^i\partial_{n-1}^j)=0
\end{array}}\\ $

\noindent This \tcb{chain complex} is denoted by $C_{\bullet}$, and elements of $C_n:=C_{(n)}$ are called \tcb{$n$-chains}, those $n$-chains $c\in C_n$ that maps to zero by the \tcb{boundary} map $d_n$ are called \tcb{$n$-cycles}.\\
\noindent \tcb{Homology groups} of the chain complex $C_{\bullet}$, are defined by $$H_n(C_{\bullet})=\Ker\ d_n/\Imm\ d_{n+1}.$$
\noindent Elements of $H_n(C_{\bullet})$ are called \tcb{homology classes}, each of which is represented by an $n$-cycle. $n$-chain that are in $\Imm \ d_{n+1}$ are called a boundaries, two $n$-chains with difference a boundary are called \tcb{homological}, even though they do not belong to $\Ker \ d_n$. \tcr{Explanation of the terminology!}\\
\subsection{Cochain Complexes}
\noindent On the other hand we can define $d^n:C_{(n)}\rightarrow C_{(n+1)}$:
$$d^n=\displaystyle\sum_{i=0}^n(-1)^iC(\sigma_n^i),$$ Which turns $$...\rightarrow C_{(n-1)}\stackrel{d^{n-1}}{\rightarrow} C_{(n)}\stackrel{d^{n}}{\rightarrow} C_{(n+1)}\rightarrow...$$ into a cochain complex that, for $n\geq 0$,\\

\noindent${ \begin{array}{ll}
d^{n+1}d^n&=\left(\displaystyle\sum_{j=0}^{n+1}(-1)^jC(\sigma_{n+1}^j)\right) \left(\displaystyle\sum_{i=0}^n(-1)^iC(\sigma_n^i)\right)=
\displaystyle\sum_{i=0}^n\displaystyle\sum_{j=0}^{n+1}(-1)^{i+j}C(\sigma_n^i\sigma_{n+1}^j)\\&=
\displaystyle\sum_{i=0}^n\displaystyle\sum_{j=0}^{i}(-1)^{i+j}C(\sigma_n^i\sigma_{n+1}^j)+\displaystyle\sum_{i=0}^{n}\displaystyle\sum_{j=i+1}^{n+1}(-1)^{i+j}C(\sigma_n^i\sigma_{n+1}^j)\\&=
\displaystyle\sum_{i=0}^n\displaystyle\sum_{j=0}^{i}(-1)^{i+j}C(\sigma_n^j\sigma_{n+1}^{i+1})+\displaystyle\sum_{i=0}^{n}\displaystyle\sum_{j=i+1}^{n+1}(-1)^{i+j}C(\sigma_n^i\sigma_{n+1}^j)\\&=
\displaystyle\sum_{k=1}^{n+1}\displaystyle\sum_{j=0}^{k-1}(-1)^{i+k+1}C(\sigma_n^j\sigma_{n+1}^{k})+\displaystyle\sum_{i=0}^{n}\displaystyle\sum_{j=i+1}^{n+1}(-1)^{i+j}C(\sigma_n^i\sigma_{n+1}^j)\\&=
\displaystyle\sum_{j=0}^{n}\displaystyle\sum_{k=j+1}^{n+1}(-1)^{i+k+1}C(\sigma_n^j\sigma_{n+1}^{k})+\displaystyle\sum_{i=0}^{n}\displaystyle\sum_{j=i+1}^{n+1}(-1)^{i+j}C(\sigma_n^i\sigma_{n+1}^j)=0
\end{array}}\\ $

\noindent This \tcb{cochain complex} is denoted by $C^{\bullet}$, and elements of $C^n:=C_{(n)}$ are called \tcb{$n$-cochains}, those $n$-cochains that maps to zero by the \tcb{coboundary} map $d^n$ are called \tcb{$n$-cocycles}.\\
\noindent \tcb{Cohomology groups} of the chain complex $C^{\bullet}$, are defined by $$H^n(C_{\bullet})=\Ker\ d^n/\Imm\ d^{n-1}.$$
\noindent Elements of $H^n(C_{\bullet})$ are called \tcb{cohomology classes}, each of which is represented by an $n$-cocycle. $n$-cochain that are in $\Imm \ d^{n-1}$ are called a boundaries, two $n$-chains with difference a boundary are called \tcb{cohomological}, even though they do not belong to $\Ker \ d^n$. 


\section{Simplicial Homotopy}
\black
\section{Algebraic Topology}
Serre Fibration, homotopy groups functor, Weak equivalence, Homotopy Equivalence, White head theorem.
























